Short Answer:
There is an official date for TOS as well as official dates for some other
Star Trek movies, television shows, and other productions.
According to that official chronology, Kirk is promoted to captain of the Enterprise in AD 2263, "where No Man Has Gone Before" happens in 2265, kirk's five year mission begins in AD 1164, season 1 of TOS happens in AD 2266 to 2268, season 2 of TOS happens in AD 2267 to 2268, season 3 of TOS happens in AD 2268 to 2269, and Kirk's five year mission ends in AD 2269.
Thus the official chronology puts TOS in the 23rd century from AD 2201 to 2300.
Long Answer:
The official chronology is not stated in canon, and thus is not canonical. Thus if could be either correct or incorrect.
The official chronology is largely based on assumptions about when various
Star Trek productions happen, instead of deducing from evidence in those productions possible date ranges for those productions. Thus it is alway s possiblity that a close examination of the evidence will prove that some
Star Trek productions cannot prossibly hppen when the official chronology says that they happen.
Those assumptions include:
The seasons of TOS happen during the overlapping years which are exactly 300 years after those seasons were brrroadcast in th e USA. Thus each TOS episode must happen 299 or 300 years after first being broadcast in the USA.
and:
Episodes of TNG happen with seasons filling Earth calendar years, with the first season of TNG happein gn 2364 exactly 400 years after the first TOS pilot was produced, the second season happening in 2365, the third in 2366, and so on.
And:
Time intervals expressed as X centuries are exactly X hundred years, no more, no less.
And:
Dates are given in the Gregorian Calendar and use
Anno Domini dating and not any other calendar eras.
https://en.wikipedia.org/wiki/Gregorian_calendar
https://en.wikipedia.org/wiki/Anno_Domini
https://en.wikipedia.org/wiki/Calendar_era#:~:text=A calendar era is the period of time,Ethiopian Orthodox churches have their own Christian eras).
I note that in all of TOS, TAS, TOS and TNG movies, and episodes of TNH, DS9, and VOY, there are hundreds of dates given, and only about half a dozen are specified as AD or BC. All other dates are totally unspecified and might be dates using other calendar eras.
And I have found considerable evidence for the use of dates given in different calendar eras in various
Star Trek productions.
And other assumptions:.
How correct are those assumptions?
In "Wolf in the Fold":
KIRK: Computer. Criminological files. Cases of unsolved mass murders of women since Jack the Ripper.
COMPUTER: Working. 1932. Shanghai, China, Earth. Seven women knifed to death. 1974, Kiev, USSR, Earth. Five women knifed to death. 2105. Martian colonies. Eight women knifed to death. 2156. Heliopolis, Alpha Eridani Two. Ten women knifed to death. There are additional examples.
So TOS would have to happen in or after AD 2156 if all the assumptions are correct.
In "Where No Man Has Gone Before":
DEHNER: Page three eighty seven.
MITCHELL: My love has wings. Slender, feathered things with grace in upswept curve and tapered tip. The Nightingale Woman, written by Phineas Tarbolde on the Canopius planet back in 1996. It's funny you picked that one, Doctor.
DEHNER: Why?
MITCHELL: That's one of the most passionate love sonnets of the past couple of centuries. How do you feel, Doctor?
In my opinion, Mitchell would only describe 1996 as being in the past couple of centuries if it was between one and two centuries earlier. Thus the date of "Where No Man Has Gone Before" should be between 2096 and 2196 of the calendar used.
So those two episodes date TOS to sometime between AD 2156 and AD 2196 according to some of the assumptions made in the official
Star Trek Chronlogy: The History of the Future. But that same chronology has "Where No Man Has Gone Before" happen in AD 2265.
According to the official chronology, "The Savage Curatain" happens in 2269 and they meet what apears to be Abraham LIncoln (1809-1865).
SCOTT: Lincoln died three centuries ago on a planet hundreds of light years away.
SPOCK: More that direction, Engineer.
So Spock seems to correct Scott about the currect direction to Earth, but doesn't correct Scott's statement about Abraham Lincoln dying 300 years ago.
According to the assumption that three centuries equals exactly 300 years, Abraham Lincoln would die in 1969, which is 160 years since he was born, and thus obviously had science ficitonal adventures different from his life in our timeline.
If
Star Trek is in an alternate universe where Abraham Lincoln improbably, but within the limits of possibility, lived to be 100 and died in 1909, Scott's statement that was three centuries ago" combined with the assumption that three centuries equals 300 years would force "The Savage Curtain" to happen in 2209 instead of 2269.
If the assumption that "three centuries" always equals exactly 300 years is abandoned, it would be possible for Abraham Lincoln to die in 1865 and "The savage Curtain" happen in 2269, if Scott was speaking loosely and vaguely enough.
I can image that Spock might make allowances for human imprecision and not correct a statement of X centuries if the actual time span was between X minus one centuries and X plus one centuries. Thus Spock woud not correct Scot about the time span if "The Savage Curtain happened between 2065 and 2265.
But the official date for "The Savage Curtain" is 2269, which is 404 years after Lincoln died in 1865. And I find it hard to believe that Spock would resist correcting Scott's "three centuries" if the actual time span was less than 200 years or more than 400 years.
So I find it very hard to believe that TOS happens in the years assigned to it by the offiicial chronology.
Thus a fictional universe which includes TOS, TAS, and the TOS films, can have TOS happen in the 22nd century or earlier in the 23rd century than the offficial chronology puts it.
But considering the many date references in TNG, DS9, and VOY which were written to agree with the official Chronology, a ficitonal universe which includes both TOS and the TNG era productions will be bound by the evidence for the time period of TNG, DS9, and VOY and for the time intervals between them and TOS and TOS era movies. Thus TOS should still happen years or decades before its official chronology date, but the connection to the TNG era productions would probably strongly limit how much earlier it could be, and probably prevent TOS from happening as early as the 22nd century AD.